NCERT Solutions Class 6 Maths Chapter 2 Lines and Angles

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 6 · M AT H S

NCERT Solutions

Chapter 2: Lines and Angles A B

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

13 – 54 21 75 English

Solutions, notes, sample papers & more at 90 pages

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

CLASS 6 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 2: Lines and Angles
Complete, step-by-step NCERT Solutions for Class 6 Maths Chapter 2 Lines and Angles from the NCERT
textbook Ganita Prakash. Every Figure it Out question from pages 15 to 54 is solved, along with all the in-text
questions, using clear diagrams, protractor readings and reasoning.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 6) 13 – 54

SECTIONS QUESTIONS

21 75

MEDIUM

English

In-text Questions — Pages 14 & 15
Sections 2.2 & 2.3 Line Segment and Line

Q1 Mark any two points A and B on a sheet of paper. Try to connect A to B by various
routes (Fig. 2.1). What is the shortest route from A to B?

B

A

Fig. 2.1 — several routes joining A to B; only one of them is straight.

The straight path from A to B is the shortest route. All the curved and bent routes are longer.

Page 1 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

This shortest path from point A to point B — including A and B — is called the line segment AB
(or BA). The points A and B are its end points.

A P Q C D O A
← → →

Point Line segment Line Ray

A point, a line segment PQ, a line CD (arrows show it never ends) and a ray OA.

Why it happens: every bend forces you to travel sideways before you can come back
towards B, and that extra sideways travel adds length. Only a path with no bends at
all wastes nothing. That is why a taut thread between two nails, or a stretched
rubber band, always becomes straight.

Try This: Join two points with a thread along a curvy route, then straighten the
thread out. Compare its length with the straight distance — the thread is always
longer.

Page 2 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q2 Imagine that the line segment from A to B (i.e., AB) is extended beyond A in one
direction and beyond B in the other direction without any end. This is a model for a
line. Do you think you can draw a complete picture of a line? No. Why?

m

B

A

Fig. 2.2 — the line m through A and B, going on endlessly both ways.

No — a complete picture of a line can never be drawn.
A line has no end points. It goes on endlessly in both directions, so however long a paper you
take, the line would still have to continue beyond both edges.

What we do instead: we draw only a part of the line and put small arrowheads at
the two ends. The arrowheads are a promise that the line keeps going. That is why a
line through A and B is written as AB with arrows above it, while the segment AB is
written without arrows.

Remember: two points are enough to fix a line completely — through any two
points there passes exactly one line.

Figure it Out — Pages 15 – 17

Page 3 of 90

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Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

Section 2.4 Ray
co m
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Rihan marked a point on a piece of paper. How many lines can he draw that pass
g
em the point? Sheetal marked two points on a piece of paper. How many
Q1

a s
through

a gl different lines can she draw that pass through both of the points? Can you help
Rihan and Sheetal find their answers?

co m
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as

a g l
Rihan can draw endlessly many lines (an uncountable number) through his single point.
Sheetal can draw only one line through her two points.

. c om
s e
Why it happens: a single point does not fix a direction. Keep the pencilm on that point
.
and turn
a la can turn it by as
m ruler a little — you get a new line every time, andgyou
cothe
a s eman amount as you like. But the moment a second point is fixed, the direction is
gl decided as well, so the line has no freedom left.
small
a
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Through 1 point → countless lines

Through 2 points → exactly 1 g l a
aline

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Check it yourself: put a ruler on two dots on your notebook. However you slide or
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turn it, you can draw only one straight line that touches both dots.
m a
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m as e
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co m
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m ase
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Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q2 Name the line segments in Fig. 2.4. Which of the five marked points are on exactly
one of the line segments? Which are on two of the line segments?

Q

M

R

P
L

Fig. 2.4 — the five marked points L, M, P, Q and R joined end to end.

Q

M

R

P

L

Fig. 2.4 — the four line segments LM, MP, PQ and QR. L and R are free ends; M, P and Q are joints.

The four line segments are:

Page 5 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

LM, MP, PQ and QR

POINT SEGMENTS IT LIES ON HOW MANY

L LM 1

M LM, MP 2

P MP, PQ 2

Q PQ, QR 2

R QR 1

On exactly one segment: L and R. On two segments: M, P and Q.

Why: L and R are the two free ends of the zig-zag path, so only one segment reaches
them. Every other point is a joint where two segments meet.

Page 6 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q3 Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays?

A

T
N B
Fig. 2.5 — two rays starting at T; N and B lie on the horizontal one, A on the slanting one.

A →

→
T N B

Page 7 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Fig. 2.5 — the rays TA, TB and TN all start at T, but the ray NB starts at N.

The rays are TA, TB, TN and NB.
No, T is not the starting point of every ray.

T is the starting point of TA, TB and TN.
The starting point of NB is N, not T.

Why the order of letters matters: in the name of a ray the first letter is always the
starting point and the second letter is any other point on it. So TB and NB run along
the same straight path, but they are different rays because they start at different
points.

Q4 Draw a rough figure and write labels appropriately to illustrate each of the
following: a. OP and OQ meet at O. b. XY and PQ intersect at point M. c. Line l
contains points E and F but not point D. d. Point P lies on AB.

Rough figures for the four situations are shown below.

Q X Q D

M l P
← →
E F A B

O P P Y
a. b. c. d.

Rough figures for the four situations (a) to (d).

a. Two rays are drawn from the same starting point O — one through P, one through Q. Their
common starting point O is where they meet.
b. Two lines XY and PQ are drawn crossing each other. The crossing point is labelled M.
c. A line is drawn and named l. Points E and F are marked on it, while D is marked away from
it.
d. A line segment AB is drawn and P is marked as a point between A and B, on the segment
itself.

Page 8 of 90

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Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

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m.
Tip: a rough figure need not be neat or to scale — but the labels must be exactly

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com on a ray. l
where the question says. Put arrowheads on a line, dots at the ends of a segment,
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and one arrowhead
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In Fig. 2.6, name: a. Five points b. A line c. Four rays d. Five line segments
.
Q5

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a

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m B glas
m .co a
ase
agl
Om a s
.co agl
C

Ease
m
agl
D
co m
m .
o m a se from O.
Fig. 2.6 — the line through D, E, O and B, with two more rays ldrawn
m .c ag
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co m
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m ase
.co


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Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Fig. 2.6 — D, E, O and B lie on one line; OC is a separate ray drawn from O.

In the figure, D, E, O and B lie on one straight line, and OC is a separate ray from O.

PART ANSWER

a. Five points D, E, O, B, C

b. A line DB (it may also be named DE, DO, EO, EB or OB)

c. Four rays OC, OB, OE, OD

d. Five line segments DE, DO, DB, EO, EB (OB and OC are also correct)

Why so many names are possible: a line can be named using any two points on it,
so the one straight line through D, E, O and B has several correct names. But a ray
must always be named with its starting point first — every ray in this figure starts at
O.

Page 10 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q6 Here is a ray OA (Fig. 2.7). It starts at O and passes through the point A. It also
passes through the point B. a. Can you also name it as OB? Why? b. Can we write OA
as AO? Why or why not?

A
B

O

Fig. 2.7 — the ray OA; it starts at O and passes through B and then A.

→
O B A

Fig. 2.7 — the single ray that starts at O and passes through B and then A. It may be called OA or OB.

a. Yes, the same ray can be named OB.
A ray is named by its starting point followed by any other point lying on it. Here the starting point
is O, and B lies on the ray just as A does. So OA and OB are two names for one and the same ray.
b. No, OA cannot be written as AO.

Page 11 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Ray OA — starts at O and goes past A, endlessly

Ray AO — starts at A and goes past O, in the opposite direction

Why it happens: unlike a line segment (where AB and BA mean the same thing), a
ray has a definite starting point and a definite direction. Changing the order of the
letters changes both. So OA ≠ AO.

In-text Questions — Pages 17 & 18
Section 2.5 Angle

Q1 Vidya has just opened her book. Let us observe her opening the cover of the book in
different scenarios (Case 1 to Case 6). Do you see angles being made in each of
these cases? Can you mark their arms and vertex?

Yes — an angle is formed in every case.

The vertex is the spine of the book, the fixed edge about which the cover turns.
The two arms are the cover of the book and the page (or table) lying flat underneath it.

D

∠DBE E

B

An angle: two rays BD and BE from the common starting point B. B is the vertex; BD and BE are the
arms.

Page 12 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Why: an angle is made by two rays that start from the same point. The spine is that
common starting point; the cover and the base sheet act like two rays going out
from it. As the cover is lifted, only the amount of turn changes — so the angle grows
while the arms stay the same length.

Q2 Which angle is greater—the angle in Case 1 or the angle in Case 2?

The angle in Case 2 is greater.
To reach Case 2, Vidya has to rotate the cover more than she did for Case 1.

Case 1 < Case 2 < Case 3 < Case 4 < Case 5 < Case 6

The big idea: just as the size of a line segment is measured by its length, the size of
an angle is measured by the amount of rotation needed about the vertex to move
the first ray onto the second ray. More turn means a bigger angle.

Page 13 of 90

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Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

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In a compass or divider, we turn the arms to form an angle... A pair of scissors has
se
Q3

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two blades... Look at the pictures of spectacles, wallet and other common objects.
g
m .c the angles in them by marking out their arms and vertices.
Identify a
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a

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spectacles wallet stapler

m as e
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se m
g l a
a
m a s
m .co agl
l a se
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pencil box

a carton

m
Everyday objects that open and shut. In each one the hinge is the vertex (red dot) and
the two parts that turn are the arms of the angle.
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m l as
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a s em
a gl

m
Every one of these objects has a hinge — and a hinge is a vertex.

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OBJECT VERTEX (FIXED ARMS OF THE ANGLE

a s
Compass / divider
atheglscrew joining the two legs the two legs

om
Scissors the rivet at the centre the two blades

. c
a s em and the side arm
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Spectacles
l
the hinge near each glass the frame

e m .Wallet / book ag the two flaps or covers
a s
the fold or the spine

agl c
.
Door the hinge on the frame the door and the wall

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Why they are all angles: in each object one part is turned with respect to the other agl
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about a fixed point. That turning is exactly what an angle measures.

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Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Figure it Out — Pages 19 – 21
Section 2.5 Angle

MATH TALK

Q1 Can you find the angles in the given pictures? Draw the rays forming any one of the
angles and name the vertex of the angle.

C
B

D
A

A bicycle and a window grille. On the bicycle the rays DA, DB and DC are drawn from the
vertex D.

Yes. In the picture of the open scissors-like shape, one clear angle is ∠BDC.
Vertex: D — the point where the two edges meet.
Arms: the rays DB and DC drawn along the two edges.

Page 15 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

D

∠DBE E

B

An angle: two rays BD and BE from the common starting point B. B is the vertex; BD and BE are the
arms.

Try This: look for angles in the other pictures too — the open beak of a bird, the
hands of a clock, the corner of a table, the branches of a tree. In each case first find
the point where the two edges meet (the vertex), then draw the two rays along the
edges.

Q2 Draw and label an angle with arms ST and SR.

The two arms are ST and SR, so both rays must start from the same point S — that makes S the
vertex.

Page 16 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

T

∠TSR
R

S

The angle ∠TSR — both arms ST and SR start from the same point S, which is therefore the vertex.

Steps:

1. Mark a point and label it S.
2. From S draw a ray and mark a point T on it.
3. From the same point S draw another ray in a different direction and mark a point R on it.
4. Draw a small curve near S between the arms and name the angle ∠TSR (or ∠RST).
Tip: the vertex letter always sits in the middle of the name — that is why it is ∠TSR
and not ∠STR.

Q3 Explain why ∠APB cannot be labelled as ∠P.

Because at the point P there is more than one angle, so the single letter P would not tell us
which one is meant.

Page 17 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

1

P B

2

Three rays from P make three different angles — so the name “∠P” would be ambiguous.

In the figure, three rays PA, PB and PC all start at P. That gives three different angles at the same
vertex:

∠APB ∠BPC ∠APC

Why the rule exists: the name ∠P only fixes the vertex; it does not fix the two arms.
Using a point on each arm — A on one arm and B on the other — makes the angle
unmistakable. We may write ∠P only when exactly one angle is formed at P.

Page 18 of 90

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Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

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Name the angles marked in the given figure.
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Q4

m l as
m .co a g
l a se
a g

co m
P e m . ag
g l as
a Q
co m
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m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
a g
T R
co m
m .
o m l a se
m .c Two angles are marked with curved arrows: one from rayaTRg to ray TQ, and one from ray
l a se TR to ray TP.

ag
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com
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m ase
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Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

P
Q

∠PTR

∠QTR
R
T

The two marked angles at the vertex T. Measured on the page, ∠PTR ≈ 104° and ∠QTR ≈ 64°.

The two marked angles are

∠PTR (or ∠RTP) and ∠QTR (or ∠RTQ)
Both have their vertex at T. The first is formed by the arms TP and TR, the second by the arms
TQ and TR.

Tip: the vertex T is written in the middle of each name. If you measure the printed
figure you will find ∠PTR ≈ 104° (obtuse) and ∠QTR ≈ 64° (acute).

Page 20 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q5 Mark any three points on your paper that are not on one line. Label them A, B, C.
Draw all possible lines going through pairs of these points. How many lines do you
get? Name them. How many angles can you name using A, B, C? Write them down,
and mark each of them with a curve as in Fig. 2.9.

Final position of ray

Amount of turn is
the size of the angle

Vertex Initial position of ray

Fig. 2.9 — the size of an angle is the amount of turn from the first ray to the second.

C

A B

Three points not on a line give 3 lines and 3 angles, one at each corner.

Number of lines: 3. They are AB, BC and CA.

Page 21 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Pairs of points: (A, B), (B, C), (C, A) → 3 lines

Number of angles: 3. One at each corner:

∠ABC (or ∠CBA) ∠BCA (or ∠ACB) ∠CAB (or ∠BAC)

Why 3 and not more: at each point exactly two lines meet, and two rays make just
one angle. Three corners × 1 angle = 3 angles. The three points must not lie on one
line — if they did, no angle would be formed at all (only a straight angle).

Page 22 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q6 Now mark any four points on your paper so that no three of them are on one line.
Label them A, B, C, D. Draw all possible lines going through pairs of these points.
How many lines do you get? Name them. How many angles can you name using A,
B, C, D? Write them all down, and mark each of them with a curve as in Fig. 2.9.

Final position of ray

Amount of turn is
the size of the angle

Vertex Initial position of ray

Fig. 2.9 — the size of an angle is the amount of turn from the first ray to the second.

A B

D C

Four points, no three on a line, give 6 lines: AB, BC, CD, DA, AC and BD.

Page 23 of 90

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Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

Number of lines: 6.
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m l as
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AB, BC, CD, DA, AC, BD
a g
l a se
a g
Number of angles: 12 — three at each of the four points.

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VERTEX RAYS FROM IT
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ag
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a
A AB, AC, AD

∠ABC, ∠CBD, ∠ABD
m
B BA, BC, BD

∠ACB, ∠BCD, ∠ACD m.co
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C CA, CB, CD

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ase
D DA, DB, DC

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Why the count works out: with 4 points you can pick a pair in 4 × 3 ÷ 2 = 6 ways, so

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there are 6 lines. At each point three lines meet; choosing 2 of those 3 rays gives 3 ×
2 ÷ 2 = 3 angles. Four vertices × 3 =m .
a s e 12 angles.

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In-text Questions — Pages 21 & 23
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m as e
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se m
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a
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co m
m .
m as e
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m a s e
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co m
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m as e
.co


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Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Section 2.6 Comparing Angles

MATH TALK

Q1 Look at these animals opening their mouths. Do you see any angles here? If yes,
mark the arms and vertex of each one. Can you arrange the angles in this picture
from smallest to largest?

snake fish chick crocodile

Four animals with their mouths open. The red dot is the vertex (the jaw hinge) and the
red lines are the arms.

Yes. Every open mouth makes an angle.

Vertex: the hinge of the jaw, where the upper and lower jaws are joined.
Arms: the upper jaw and the lower jaw.

To arrange them, look at how much each jaw has been turned: the wider the mouth is open, the
greater the turn and the larger the angle. Trace the angles on tracing paper and place them one
over the other to be sure — the widest gape goes last in the list.

Careful: a crocodile's jaw is long and a bird's beak is short, but length has nothing to
do with it. A small bird with a wide-open beak can make a bigger angle than a big
animal with a slightly open mouth.

Q2 Is it always easy to compare two angles?

No. Comparing is easy only when the two angles are clearly different.

A 30° angle and a 120° angle can be told apart at a glance.

Page 25 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

But an 89° angle and a 91° angle look almost the same — the eye cannot decide.

A

X

Y B
O O

Two angles of nearly the same size, drawn with arms of different length. The eye alone cannot decide
which is bigger.

What to do then: use one of two reliable methods — superimposition (trace one
angle and place it exactly over the other, vertex on vertex) or measurement with a
protractor. Long arms can also trick the eye into thinking an angle is bigger, which is
one more reason not to trust a quick look.

Q3 Here are some angles. Label each of the angles. How will you compare them? Draw
a few more angles; label them and compare.

First give every angle a name using three letters, with the vertex in the middle — for example
∠ABC, ∠PQR, ∠XYZ.
Then compare them in any of these ways:

1. By looking — works only when the difference is large.
2. By superimposition — trace one angle on tracing paper, place its vertex exactly on the other
vertex and one arm along the other arm, then see which second arm lies inside.
3. Using a circle — place a transparent circle with its centre on the vertex and see how much of
the circle each angle covers.
4. By measuring — read both angles with a protractor and compare the numbers. This is the
surest method.

Page 26 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Math Talk: draw four angles of your own, label them, first guess the order from
smallest to largest, then check by measuring. How many did you get right?

Q4 Where else do we use superimposition to compare?

Superimposition — placing one thing exactly over another — is used to compare many shapes
and sizes:

Line segments — put one over the other with the left ends together and see which one
sticks out.
Circles — place them centre on centre; the one whose boundary lies outside is bigger.
Squares, triangles and other figures — to check whether two cut-outs are of exactly the
same size and shape.
Leaves, footprints, bangles, coins — placed one over the other to compare size.
Tracing a map or a drawing — to check that a copy matches the original.

Did you know? Two figures that fit exactly on each other by superimposition are
called congruent figures. You will study them in higher classes.

Figure it Out — Page 23
Section 2.6 Comparing Angles

MATH TALK

Q1 Fold a rectangular sheet of paper, then draw a line along the fold created. Name
and compare the angles formed between the fold and the sides of the paper. Make
different angles by folding a rectangular sheet of paper and compare the angles.
Which is the largest and smallest angle you made?

Fold the sheet so that the crease cuts across two opposite sides. Mark the crease as EF, with E
and F on the two sides. Four angles are formed:

∠AEF, ∠BEF, ∠DFE, ∠CFE
On comparing (by tracing and superimposing them):

Page 27 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

If the crease is slanting, then ∠AEF and ∠CFE are the two obtuse angles, and ∠BEF and
∠DFE are the two acute angles. So ∠AEF = ∠CFE > ∠BEF = ∠DFE.
If the crease is folded so that it runs straight across, all four angles become equal right
angles.

∠AEF + ∠BEF = 180° (they make a straight angle along the edge)

Why: the two angles on one side of the paper together make a straight line, so
making one of them larger automatically makes the other smaller. The largest angle
you can make this way is just below a straight angle (180°), and the smallest is a very
thin sliver near 0°.

Q2 In each case, determine which angle is greater and why. a. ∠AOB or ∠XOY b. ∠AOB
or ∠XOB c. ∠XOB or ∠XOC. Discuss with your friends on how you decided which one
is greater.

X
A

Y

32°

B C
O

The rays OA, OX, OY and OB from the vertex O. B and C lie on the same ray, so OB and OC are one
arm. Measured: ∠AOB ≈ 90°, ∠XOB ≈ 60°, ∠XOY ≈ 32°.

a. ∠AOB is greater.

Page 28 of 90

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Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

co m
e m.
∠AOB = ∠AOX + ∠XOY + ∠YOB
m l as
.co a g
a s em
The whole is bigger than any of its parts, and ∠XOY is only a part of ∠AOB. Measured with a

agl
protractor, ∠AOB ≈ 90° while ∠XOY ≈ 32°.
b. ∠AOB is greater.

. com ag
∠AOB = ∠AOX + ∠XOB so ∠AOB
a s em> ∠XOB
(≈ 90° against ≈ 60°) agl

co m
c. Neither — they are equal: ∠XOB = ∠XOC.
se m.
o m l a
gfrom O. So OB and
m .c surprises many students: B and C lie on the same ray
a
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Why (c)

g l a
a
OC are two names for one and the same arm, and both angles are made by exactly
the same pair of arms. Making an arm longer does not make the angle bigger —

m a s
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only turning does.

a s em
agl
Q3 Which angle is greater: ∠XOY or ∠AOB? Give reasons.

co m
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a g
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∠XOY ≈ 111° com
∠AOB ≈ 55° e m .
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B
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a

co m
m .
m ase
.co


a g l Page 29 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

All four rays start at O. Measured on the page, ∠XOY ≈ 111° and ∠AOB ≈ 55° — but neither angle lies
inside the other.

∠XOY is the greater angle. Measuring the printed figure with a protractor gives

∠XOY ≈ 111° ∠AOB ≈ 55°

But you cannot get this answer just by looking, and the trick used in question 2 does not work
here either.

Why the “part of the whole” argument fails: in question 2 one angle lay
completely inside the other. Here the four rays are interleaved — going anticlockwise
from OB the order is B, Y, A, X. So ∠AOB sticks out below OY while ∠XOY sticks out
above OA: neither angle contains the other. Nothing can be concluded by inspection
alone.

What to do instead:

1. Superimposition — trace ∠AOB on tracing paper, put the traced vertex on O with OA along
OX, and see where OB falls.
2. Measurement — read both angles with a protractor and compare the numbers. This settles
it at once.

In-text Questions — Pages 24 – 27
Sections 2.6 & 2.7 Comparing Angles, Rotating Arms

Q1 Suppose we have a transparent circle which can be moved and placed on figures...
Can you now tell which angle is bigger? Which crane was making the bigger angle?

Yes — the circle settles the argument.
Place the transparent circle so that its centre sits exactly on the vertex of the first crane's
angle, and mark the points A and B where the two arms cross the circle. Now move the circle to
the second crane's angle, again centre on vertex, with one arm along OA.

If the second arm OY falls outside OB, the second angle is bigger.
If it falls inside OB, the second angle is smaller.
If it lies exactly on OB, the two angles are equal.

In the picture the second crane's arm falls outside, so the second crane was making the
bigger angle.

Page 30 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Why this works: the two angles are now compared on the same circle, so the arc cut
off by the arms is a fair measure of the turn. The lengths of the beaks no longer
matter at all. This is exactly the idea that leads to the protractor.

Q2 Passing through a slit: Now, shuffle and mix up all the rotating arms. Can you
identify which of the rotating arms will pass through the slit?

Only the rotating arm whose angle is exactly equal to the angle of the slit will pass
through.

SITUATION WILL IT PASS?

Slit angle greater than the arms' angle No

Slit angle less than the arms' angle No

Slit angle equal to the arms' angle Yes

Why it happens: the slit is a copy of one particular angle. Sliding the arms into it is
the same as superimposing them, and only equal angles superimpose perfectly.
Note that whether the arms pass depends only on the angle, not on the length of
the straws — as long as they are shorter than the slit.

Q3 Challenge: Reduce this angle. (The arms are cut shorter — is the angle reduced?)

No! Cutting the arms shorter does not reduce the angle — the angle is still the same.

Why it happens: the size of an angle is the amount of turn between the two arms
about the vertex. Snipping the ends of the straws changes their length, but not the
turn. The only way to reduce the angle is to rotate one arm closer to the other.

Check it yourself: draw an angle, measure it, then rub out the outer half of both
arms and measure again. The reading does not change.

Page 31 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

In-text Questions — Pages 28 – 31
Section 2.8 Special Types of Angles

MATH TALK

Q1 Let us consider a straight angle ∠AOB. Observe that any ray OC divides it into two
angles, ∠AOC and ∠COB. Is it possible to draw OC such that the two angles are
equal to each other in size? Justify why the two angles are equal.

Yes, it is possible — and paper folding shows it beautifully.
Take a rectangular sheet, mark the straight angle ∠AOB along one edge, and fold the paper so
that ray OB falls exactly on ray OA. The crease that passes through O is the required ray OC.

C

90° 90°

A B
O

The ray OC divides the straight angle ∠AOB into two equal right angles of 90° each.

Justification: when the paper is folded, ∠COB comes to lie exactly on top of ∠AOC — vertex on
vertex and arm on arm. By superimposition the two angles are therefore equal.

∠AOC + ∠COB = 180° and ∠AOC = ∠COB

so each angle = 180° ÷ 2 = 90°

What we have just made: each of these two equal angles is a right angle. So a
straight angle contains exactly two right angles, and OC is perpendicular to AB.

Page 32 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q2 If a straight angle is formed by half of a full turn, how much of a full turn will form a
right angle?

A quarter of a full turn.

1 straight angle = ½ of a full turn

1 straight angle = 2 right angles

1 right angle = ½ × ½ = ¼ of a full turn

In degrees:

Full turn = 360° → straight angle = 180° → right angle = 90°

Tip: a right angle looks like the letter L, and two lines meeting at a right angle are
called perpendicular lines. But be careful — a shape that merely looks like an L is a
right angle only if it is exactly half of a straight angle.

Q3 Angles are classified in three groups as shown. Right angles are shown in the
second group. What could be the common feature of the other two groups?

First group Second group Third group

The three groups of angles. The middle group holds the right angles.

The groups are decided by comparing each angle with a right angle.

Page 33 of 90

as e
Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

co m
m.
GROUP COMMON FEATURE NAME

m as e
First
.co
every angle is smaller than a right angle — less than a quarter turn
a g l Acute

m
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angles

gl
aSecond every angle is exactly a quarter turn Right angles

m
.co ag
Third every angle is greater than a right angle but smaller than a straight angle — between a Obtuse

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quarter turn and a half turn angles

a
agl

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m l as
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45° 90°
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130° 180° 250°
a s
acute right
m .coobtuse straight reflex
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The five kinds of angles — acute, right, obtuse, straight and reflex.

co m
Why the right angle is the yardstick: it is easy to make (fold a paper twice) and
m .
as e
com"equal to" or "more than" a right angle. l
easy to recognise (the corner of a book). So every other angle is described as "less
. a g
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than",

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Figure it Out — Pages 29 – 31
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Section 2.8 Right Angles and Perpendicular Lines

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Q1 How many right angles do the windows of your classroom contain? Do you see
other right angles in your classroom?
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1 window pane = 4 right angles
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A window with 4 panes = 4 ×e4m= 16 right angles
g l as
(plus the 4 of the outeraframe)

co m
m .
m as e
.co


a g l Page 34 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Count the panes in your own window and multiply by 4.
Other right angles in the classroom:

corners of the blackboard, the door, the notebook, the desk top and the floor tiles;
the corner where two walls meet, and where a wall meets the floor;
the corner of a chart paper, a duster, a cupboard and the almirah shelves;
the two edges of a set-square, and the corner of your geometry box.

Check it yourself: fold a paper twice to make a right-angle tester, and hold it against
these corners.

Q2 Join A to other grid points in the figure by a straight line to get a straight angle.
What are all the different ways of doing it?

A B A

B

The two dot grids: in each one the segment AB is already drawn and A has to be joined
to another grid point.

A straight angle at A means the two arms must go in exactly opposite directions from A — so
the point you join must lie on the ray AB extended backwards through A.

Page 35 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

180° 180°
A B A

B

A straight angle at A: join A to a grid point on the ray opposite to AB — straight left in the first grid,
and up-left along the diagonal in the second.

GRID STEP FROM A WHERE TO JOIN A HOW MANY SUCH
TO B GRID POINTS

First (AB 3 steps to the right any grid point of the same row to the 3
horizontal) left of A

Second (AB 2 steps right and 2 up-left along the same slant — 1 left 2
slanting) steps down & 1 up, or 2 left & 2 up

In every case the angle formed is 180°.

How many different ways? Only one direction works in each grid — the direction
exactly opposite to AB. All the grid points lying that way give the same straight angle,
because they all lie on one and the same ray. So there are several points to choose
from, but only one straight angle.

Page 36 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q3 Now join A to other grid points in the figure by a straight line to get a right angle.
What are all the different ways of doing it? (Hint: Extend the line further. To get a
right angle at A, we need to draw a line through it that divides the straight angle
CAB into two equal parts.)

A B A

B

C

A

B

The two grids for the question, and below them the hint grid in which AB is extended
back through A to C.

For a right angle at A the new line must be perpendicular to AB. On a square grid there is a
neat trick for this.

If B is reached from A by a steps right and b steps down,

then a perpendicular direction is b steps right and a steps up

(or b steps left and a steps down)

Page 37 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

90°
A B A
90°

B

A right angle at A: in the first grid go straight up or straight down; in the second go 2 left and 2 down,
or 2 right and 2 up.

GRID STEP FROM A PERPENDICULAR STEPS FROM DIFFERENT
TO B A WAYS

First (AB 3 right straight up (2 points) or straight down (3 2
horizontal) points)

Second (AB 2 right, 2 down 2 right & 2 up, or 2 left & 2 down 2
slanting)

So in each grid there are two different right angles at A — one on each side of the line AB —
and each may be drawn to any grid point along that direction.

Check it yourself: use the hint — extend BA beyond A to C. The line you draw must
cut the straight angle ∠CAB into two equal halves, so each half is 180° ÷ 2 = 90°.

Page 38 of 90

as e
Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

co m
m.
Get a slanting crease on the paper. Now, try to get another crease that is
se
Q4

o m l a
perpendicular to the slanting crease. a. How many right angles do you have now?
g folded the paper
m .cwhy the angles are exact right angles. b. Describe howayou
Justify

l a sethat any other person who doesn't know the process can simply follow your
so
a g description to get the right angle.

co m
m . ag

as e
a g l

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
ag 4 right angles

co m
m .
m ase
.co a g l
m
Two creases crossing at right angles — four exact right angles of 90° each.

l a se
a g a. There are 4 right angles — the two creases cross each other and cut the whole turn around
that point into four equal parts.
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Full turn = 360°
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4 equal angles → each = 360° ÷ 4 = 90°

. c om
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g follow:
b. Instructions anyoneacan

co m
m .
m ase
.co


a g l Page 39 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

1. Take a sheet of paper and fold it once in any slanting direction. Press the fold hard and open
it — you get a slanting crease.
2. Now fold the paper a second time, this time bringing one part of the first crease exactly
on top of the other part of the same crease.
3. Press the second fold and open the paper. The two creases meet at a point and form four
right angles.

Tip: this is the quickest way to make a perfect 90° without a protractor — the paper
does the measuring for you.

Figure it Out — Pages 31 & 32
Section 2.8 Classifying Angles

Q1 Identify acute, right, obtuse and straight angles in the previous figures.

The three groups of angles printed just above this exercise (page 31).

Compare every angle with a right angle (use the corner of your notebook as a tester):

Page 40 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

45° 90° 130° 180° 250°
acute right obtuse straight reflex

The five kinds of angles — acute, right, obtuse, straight and reflex.

TYPE HOW TO SPOT IT WHERE IT APPEARS IN THE EARLIER FIGURES

Acute smaller than the notebook the narrow angles of the rotating arms; Vidya's book in Case 1 and
corner Case 2; the thin angles at a paper fold

Right fits the notebook corner the four angles at the crossing creases; corners of the rectangular
exactly sheet; Vidya's book opened as an L

Obtuse bigger than the corner but not the wide angles at a slanting crease; Vidya's book in Case 5
a straight line

Straight the two arms lie in one ∠AOB when the book is opened flat on the table (Case 6)
straight line

Q2 Make a few acute angles and a few obtuse angles. Draw them in different
orientations.

Draw them opening upwards, downwards, to the left and to the right — the direction in which
an angle points does not change its type.

Page 41 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

45° 60° 80° 100° 135° 160°

Acute angles (45°, 60°, 80°) and obtuse angles (100°, 135°, 160°) drawn in different orientations.

Acute angles (less than 90°): try 20°, 45°, 60° and 80°.
Obtuse angles (between 90° and 180°): try 100°, 120°, 135° and 160°.

Check it yourself: hold the corner of your notebook at the vertex of each angle you
drew. If your angle hides inside the corner it is acute; if it opens out beyond the
corner it is obtuse.

Q3 Do you know what the words acute and obtuse mean? Acute means sharp and
obtuse means blunt. Why do you think these words have been chosen?

Because the words describe exactly how the angles look.

An acute angle has a narrow opening, so the vertex ends in a sharp point — like the tip of a
pencil, a needle or a thorn.
An obtuse angle has a wide opening, so the vertex looks flat and blunt — like the edge of a
spoon or a rounded stone.

45° 90° 130° 180° 250°
acute right obtuse straight reflex

Page 42 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

The five kinds of angles — acute, right, obtuse, straight and reflex.

The idea: the smaller the turn between the arms, the sharper the corner appears;
the larger the turn, the more the corner opens out and flattens. So "sharp" and
"blunt" are good everyday names for "less than 90°" and "more than 90°".

Did you know? Doctors also use these words — an acute illness is sharp and
sudden, while a blunt object causes an obtuse injury.

Q4 Find out the number of acute angles in each of the figures below. What will be the
next figure and how many acute angles will it have? Do you notice any pattern in
the numbers?

(i) (ii) (iii)

The three figures: a plain triangle, then one and then two more triangles drawn inside
it.

3 acute angles 12 acute angles 21 acute angles

Page 43 of 90

as e
Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

The three figures of page 32 — 3, 12 and 21 acute angles. Each new inner triangle turns one small
co m
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triangle into four.

m l as
m .co a g
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FIGURE
a se SMALL TRIANGLES IN IT NUMBER OF ACUTE ANGLES
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The next figure is made by joining the midpoints of one more small triangle, and it contains 30
. a g
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acute angles.

a s
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The pattern:

m a s
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3, 3 + 9 = 12, 12 + 9 = 21, 21 + 9 = 30, 30 + 9 = 39 …

se m
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Why 9 more each time: every triangle in these pictures is equilateral, so each of its
three angles is 60° — acute. Joining the midpoints of one small triangle replaces it by

co m
4 smaller ones, that is 3 extra triangles, and each brings 3 more acute angles:
m .
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In-text Questions — Page 34
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Section 2.9 Measuring Angles
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s e mQ1 The circle has been divided into 1, 2, 3, 4, 5, 6, 8, 9, 10 and 12 parts below. What are

agla the degree measures of the resulting angles? Write the degree measures down near

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m a s e
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A full turn is 360°. So to find each angle, simply divide 360 by the number of equal parts.

co m
m .
m ase
.co


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Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

NUMBER OF EQUAL PARTS WORKING ANGLE OF EACH PART

1 360 ÷ 1 360°

2 360 ÷ 2 180°

3 360 ÷ 3 120°

4 360 ÷ 4 90°

5 360 ÷ 5 72°

6 360 ÷ 6 60°

8 360 ÷ 8 45°

9 360 ÷ 9 40°

10 360 ÷ 10 36°

12 360 ÷ 12 30°

Why 360 was chosen: 360 can be divided exactly by every number up to 10 except 7
— and also by 12 (the months of a year) and 24 (the hours of a day). That is why each
of the divisions above gives a whole number of degrees. The Rigveda too speaks of a
wheel with 360 spokes.

Did you know? 360 ÷ 7 = 51.43… is not a whole number, which is exactly why the
circle is never divided into 7 equal parts in this list.

Figure it Out — Page 35

Page 45 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Section 2.9 The Unlabelled Protractor

Q1 Write the measures of the following angles: a. ∠KAL b. ∠WAL c. ∠TAK. (Notice that
the vertex of this angle coincides with the centre of the protractor. So the number
of units of 1 degree angle between KA and AL gives the measure of ∠KAL. By
counting, we get ∠KAL = 30°. Making use of the medium sized and large sized
marks, is it possible to count the number of units in 5s or 10s?)

Page 46 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

K

L

A

W

T

The unlabelled protractor of page 35. Counting the long (10°) marks gives ∠KAL = 30°, ∠WAL = 50°
and ∠TAK = 120°.

Count the 1° units between the two arms. The long marks come after every 10° and the medium
marks after every 5°, so you can count in tens and fives instead of one by one.

Page 47 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

ANGLE HOW TO COUNT MEASURE

a. ∠KAL 3 long marks = 3 × 10 30°

b. ∠WAL 5 long marks = 5 × 10 50°

c. ∠TAK 12 long marks = 12 × 10 120°

Yes, counting in 5s and 10s is definitely possible — and it is far quicker and safer than
counting 120 single marks.

Check it yourself: measured on the printed page the three rays sit at 30.5°, 52.4°
and 120.3° from one another — exactly the 30°, 50° and 120° that careful counting
gives.

Why the marks are of three sizes: long marks every 10°, medium marks every 5°
and short marks every 1°. They act like the big and small divisions on a ruler, so the
eye can jump ten at a time and then adjust.

In-text Questions — Pages 36 & 37
Section 2.9 The Labelled Protractor

Q1 There are two sets of numbers on the protractor: one increasing from right to left
and the other increasing from left to right. Why does it include two sets of
numbers?

So that the protractor can be used from either side without any subtraction.

If the base arm of your angle points to the right, start at the 0 on the right and read the
inner set of numbers.
If the base arm points to the left, start at the 0 on the left and read the outer set.

Inner reading + outer reading = 180° at every mark

(for example 20 and 160, 35 and 145, 90 and 90)

Page 48 of 90

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Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

co m
m.
Why it helps: without the second set you would have to turn the paper around, or

m as e
l
measure one number and subtract it from 180 every time. The two scales save that
step.
m .co a g
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aTip: always check first which arm lies on the 0 mark, then follow that same scale to

co m
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the other arm. Mixing the two scales is the most common protractor mistake.
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m a s e
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g l as
a

co m
m .
m ase
.co


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Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q2 Name the different angles in the figure and write their measures. Did you include
angles such as ∠TOQ? Which set of markings did you use — inner or outer?

R
S

70
80 90 100
110 Q
12
60 0
100 90 80 13
110 70
50 60
0
0
12
0 50

14
13
T 40

0
0

40

15
14
30

0
0

30
15

160
20

160

20

170
10

180 170

10
U P

180
0

0
O

A protractor with its centre at O; the rays OP, OQ, OR, OS, OT and OU pass through the
0°, 35°, 95°, 125°, 160° and 180° marks of the inner scale.

R

S
100 80
120 80 100 60
60 120 Q
140 40
40 140
T
160 20
20 160

U 180 0 180 0 P
O

The six rays read on one scale: P at 0°, Q at 35°, R at 95°, S at 125°, T at 160° and U at 180°.

Page 50 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

The rays OP, OQ, OR, OS, OT and OU start from the centre O. Reading from the 0 mark at P (that
is, using one single scale throughout):

P → 0° Q → 35° R → 95° S → 125° T → 160° U → 180°

Every angle is now just a subtraction of two readings:

ANGLE WORKING MEASURE

∠POQ 35 − 0 35°

∠POR 95 − 0 95°

∠POS 125 − 0 125°

∠POT 160 − 0 160°

∠POU 180 − 0 180° (straight angle)

∠QOR 95 − 35 60°

∠QOS 125 − 35 90°

∠QOT 160 − 35 125°

∠QOU 180 − 35 145°

∠ROS 125 − 95 30°

∠ROT 160 − 95 65°

∠ROU 180 − 95 85°

∠SOT 160 − 125 35°

∠SOU 180 − 125 55°

∠TOU 180 − 160 20°

Yes, angles such as ∠TOQ must also be included — it is the same as ∠QOT = 125°. In all, 15
angles can be named from these six rays.

Tip: use one set of markings — inner or outer — for the whole figure. Here the outer
scale is convenient because OT and OS fall on the round numbers 20 and 55 on it.

Page 51 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q3 What is the measure of ∠TOS? Can you use the numbers marked to find the angle
without counting the number of markings? Here, OT and OS pass through the
numbers 20 and 55 on the outer scale. How many units of 1 degree are contained
between these two arms? Can subtraction be used here?

Yes — subtraction is exactly the short cut we need.

OS is at 55 OT is at 20 (outer scale)
∠TOS = 55 − 20 = 35°

So there are 35 units of 1° between the two arms, and we did not have to count a single mark.

Why subtraction works: the reading of an arm tells us how far that arm has turned
from the 0 mark. If one arm has turned 20 units and the other 55 units from the
same starting line, the turn between them is the difference, 55 − 20.

Check it yourself: read the same two arms on the inner scale — they are at 160 and
125, and 160 − 125 = 35°. The same answer, as it must be.

Q4 How can we measure angles directly without having to subtract? Place the
protractor so the centre is on the vertex of the angle. Align the protractor so that
one of the arms passes through the 0º mark. What is the degree measure of ∠AOB?

If one arm is placed on the 0 mark, then the reading of the other arm is the answer — the
subtraction becomes 0 and disappears.

Reading of OB = 0

Reading of OA = 80

∠AOB = 80 − 0 = 80°

The three steps to remember:

1. Centre on vertex — put the mid-point of the protractor's base exactly on O.
2. Base on one arm — turn the protractor until OB lies along the 0 line.
3. Read the other arm — follow the scale that starts from 0 on OB up to OA.

Page 52 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Check it yourself: measured on the printed figure the ray OA stands at 80.3° above
OB, so 80° is the reading the picture is meant to give. If an arm is too short to reach
the scale, extend it with a ruler first — making an arm longer never changes the
angle.

In-text Questions — Pages 38 – 40
Section 2.9 Make your own Protractor

TRY THIS

Q1 The measure of half a circle is ½ of a full turn. So, the measure of half a turn = ½ of
____ = 180°. The measure of a ¼ turn = ¼ of 360° = ________. When folded, this is ⅛ of
the circle, or ⅛ of a turn, or ⅛ of 360°, or ¼ of 180° or ½ of 90° = ________. Continuing
with another half fold, we get an angle of measure ________.

Each fold halves the angle. Fill the blanks like this:

FOLD PART OF A FULL TURN WORKING ANGLE

Whole circle 1 — 360°

1st fold (semicircle) ½ ½ of 360° 180°

2nd fold (quarter) ¼ ¼ of 360° = ½ of 180° 90°

3rd fold ⅛ ⅛ of 360° = ¼ of 180° = ½ of 90° 45°

4th fold 1/16 ½ of 45° 22.5°

So the blanks are 360°, 90°, 45° and 22.5°. The new creases also give 180° − 45° = 135°, and later
67.5°, 112.5° and 157.5°.

Tip: write 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5° and 180° along the edge of
your paper protractor — it can then measure any of these angles without a scale.

Page 53 of 90

as e
Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG . com
= ∠GOH
m
Think! In Fig. 2.19, we have
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Q2
∠HOI = _____. Why? a
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as
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Fig. 2.19 — the semicircle folded into eight equal parts, marked O, A, B, … , I.
a
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Each of these angles is 22.5°.

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s m straight angle ∠AOI = 180°
eThe
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It has been folded into 8 equal parts

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Each part = 180° ÷ 8 = 22.5°
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Why they are all equal: every crease was made by folding the previous angle

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exactly in half, so at each stage the two new angles fell exactly on each other. Halving
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180° three times gives 90°, then 45°, then 22.5° — and the eight small angles
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produced are copies of one another.
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Check it yourself: 22.5° × 8 = 180° ✔ and 22.5° × 16 = 360°, one full turn.

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a g l Page 54 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q3 Identify the angle bisectors in your handmade protractor. Try to make different
angles using the concept of angle bisector through paper folding.

A line that cuts an angle into two equal parts is called the angle bisector of that angle. In the
paper protractor every crease is a bisector of the angle formed by the two creases beside it.

CREASE AT BISECTS THE ANGLE INTO TWO ANGLES OF

90° the straight angle 0°–180° 90° each

45° the right angle 0°–90° 45° each

135° the right angle 90°–180° 45° each

22.5° the angle 0°–45° 22.5° each

67.5° the angle 45°–90° 22.5° each

Try This: fold once more to get 11.25°, and combine creases to build angles such as
22.5 + 45 = 67.5°, or 90 + 22.5 = 112.5°. Paper folding alone can make a surprising
number of exact angles.

Figure it Out — Pages 40 – 43
Section 2.9 Measuring Angles with a Protractor

Q1 Find the degree measures of the following angles using your protractor.

Place the centre of the protractor on the vertex H, put one arm on the 0 line, and read the other
arm on the same scale.

Page 55 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

108°
47°
23°

The three angles of page 40 — 47°, 23° and 108°, drawn in different directions.

FIGURE ANGLE MEASURE TYPE

1st (vertex H at the bottom) ∠IHJ 47° acute

2nd (vertex H at the top) ∠GHK 23° acute

3rd (vertex H on the left) ∠IHJ 108° obtuse

In the middle figure I and J lie on the very same two rays as G and K, so that angle may equally
well be named ∠IHJ.
Tip: if an arm is too short to reach the curved scale, extend it with your ruler. The
angle does not change when the arms are made longer. (Measured exactly, the three
angles are 47.0°, 23.5° and 108.8°.)

Q2 Find the degree measures of different angles in your classroom using your
protractor.

This is an activity — measure and record. Here is a table you can copy and fill in:

Page 56 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

OBJECT WHERE THE ANGLE IS EXPECTED MEASURE

Notebook / blackboard corner between two edges 90°

Open door between the door and the wall anything from 0° to about 180°

Open scissors between the blades an acute angle, say 30°–60°

Hands of the wall clock at the centre a multiple of 30° at exact hours

Set-square its three corners 90°, 60°, 30° (or 90°, 45°, 45°)

Open geometry box lid between lid and box an obtuse angle

Check it yourself: measure the three corners of your set-square and add them. You
should get 180° — the same result as question 9 below.

Q3 Find the degree measures for the angles given below. Check if your paper
protractor can be used here!

First angle: ∠IHJ = 42° (acute)

Second angle: ∠IHJ = 116° (obtuse)

(Measured exactly on the printed page they are 41.7° and 116.3°.)
No, the paper protractor cannot be used here.

Why not: the handmade paper protractor was built only by repeated halving, so it
carries just the marks 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5° and 180°. The
angles 42° and 116° do not fall on any of those creases. A standard protractor is
needed because it is divided into all 180 single degrees.

Tip: the paper protractor can still tell you a lot — 42° lies between 22.5° and 45° (so it
is acute), and 116° lies between 112.5° and 135° (so it is obtuse).

Page 57 of 90

Class 6 Maths Chapter 2 Lines and Angles AglaSem · NCERT Solutions

Q4 How can you find the degree measure of the angle given below using a protractor?

The marked angle is a reflex angle, bigger than 180°, so it cannot be read directly — a
protractor only goes up to 180°.
Method: measure the other angle at the same vertex (the unmarked one), then subtract it from
a full turn.

Marked angle = 360° − unmarked angle

= 360° − 101° = 259°

100°

A
O

260°

The marked reflex angle: 360° − 101° = 259°.

Why it works: the marked angle and the unmarked angle together make one
complete turn about the vertex, and a complete turn is 360°.

Note: the unmarked angle of the printed figure measures 101.5°, which gives a
reflex angle of about 258°. The textbook's answer key rounds the reading to 100°
and so gives 260°. Any answer between about 258° and 260° is right — what matters
is the method.

Page 58 of 90

as e
Class 6 Maths Chapter 2 Lines and Angles
a g l AglaSem · NCERT Solutions

co m
m.
Another way: split the reflex angle into a straight angle plus the rest — 180° + 78° ≈

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l
258° — and add the two parts.

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l a se
a g
Q5 Measure and write the degree measures for each of the following angles: a, b, c, d,

com
ag
e, f.

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Place the centre of the protractor on each vertex, one arm on 0, and read off the other arm:

co m
ANGLE MEASURE
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TYPE
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a. 80°

a
aglb. 120° obtuse

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c. 60° acute

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d. 130° obtuse

e.
g
a130° obtuse

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f. acute
m .
60°

o m l a se
m .c it yourself: measured very precisely on the printedagpage the six angles come
aseto 79°, 120.5°, 59.5°, 129°, 128° and 61°. So a reading within a degree or two of the
Check

agl
m
values above is perfectly correct.

a se
. com a g l
m
ase
Tip: angles (d) and (e) are equal even though they are drawn in different directions

agl
and with arms of different lengths — turn your protractor, not your judgement. The
same is true of (c) and (f).

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m .
m as e
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se m
g l a
a c
m .
m a s e
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a

co m
m .
m ase
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a g l Page 59 of 90